Geometry of differential operators, odd Laplacians, and homotopy algebras

We give a complete description of differential operators generating a given bracket. In particular we consider the case of Jacobi-type identities for odd operators and brackets. This is related with homotopy algebras using the derived bracket construction. (Based on a talk at XXII Workshop on Geometric Methods in Physics at Bialowieza)Comment: 13 pages; LaTe

Supersymmetry and the Odd Poisson Bracket

Some applications of the odd Poisson bracket developed by Kharkov's theorists are represented, including the reformulation of classical Hamiltonian dynamics, the description of hydrodynamics as a Hamilton system by means of the odd bracket and the dynamics formulation with the Grassmann-odd Lagrangian. Quantum representations of the odd bracket are also constructed and applied for the quantization of classical systems based on the odd bracket and for the realization of the idea of a composite spinor structure of space-time. At last, the linear odd bracket, corresponding to a semi-simple Lie group, is introduced on the Grassmann algebra.Comment: 17 pages, LATEX 2e. Invited talk given at the International Symposium "30 Years of Supersymmetry" (Theoretical Physics Institute, University of Minnesota, Minneapolis, MN, USA, 13-27 October, 2000) due to the support kindly offered by the Organizing Committee of this meeting and, especially, by Keith Olive and Mikhail Shifma

On odd Laplace operators

We consider odd Laplace operators acting on densities of various weight on an odd Poisson (= Schouten) manifold $M$. We prove that the case of densities of weight 1/2 (half-densities) is distinguished by the existence of a unique odd Laplace operator depending only on a point of an ``orbit space'' of volume forms. This includes earlier results for odd symplectic case, where there is a canonical odd Laplacian on half-densities. The space of volume forms on $M$ is partitioned into orbits by a natural groupoid whose arrows correspond to the solutions of the quantum Batalin--Vilkovisky equations. We give a comparison with the situation for Riemannian and even Poisson manifolds. In particular, the square of odd Laplace operator happens to be a Poisson vector field defining an analog of Weinstein's ``modular class''.Comment: LaTeX2e, 18p. Exposition reworked and slightly compressed; we added a table with a comparison of odd Poisson geometry with Riemannian and even Poisson cases. Latest update: minor editin

Port-Hamiltonian systems: an introductory survey

The theory of port-Hamiltonian systems provides a framework for the geometric description of network models of physical systems. It turns out that port-based network models of physical systems immediately lend themselves to a Hamiltonian description. While the usual geometric approach to Hamiltonian systems is based on the canonical symplectic structure of the phase space or on a Poisson structure that is obtained by (symmetry) reduction of the phase space, in the case of a port-Hamiltonian system the geometric structure derives from the interconnection of its sub-systems. This motivates to consider Dirac structures instead of Poisson structures, since this notion enables one to define Hamiltonian systems with algebraic constraints. As a result, any power-conserving interconnection of port-Hamiltonian systems again defines a port-Hamiltonian system. The port-Hamiltonian description offers a systematic framework for analysis, control and simulation of complex physical systems, for lumped-parameter as well as for distributed-parameter models

ARPES Spectral Function in Lightly Doped and Antiferromagnetically Ordered YBa2Cu3O{6+y}

At doping below 6% the bilayer cuprate YBa2Cu3O{6+y} is a collinear antiferromagnet. Independent of doping the value of the staggered magnetization at zero temperature is about 0.6\mu_B. This is the maximum value of the magnetization allowed by quantum fluctuations of localized spins. In this low doping regime the compound is a normal conductor with a finite resistivity at zero temperature. These experimental observations create a unique opportunity for theory to perform a controlled calculation of the electron spectral function. In the present work we perform this calculation within the framework of the extended t-J model. As one expects the Fermi surface consists of small hole pockets centered at (\pi/2,\pi/2). The electron spectral function is very strongly anisotropic with maximum of intensity located at the inner parts of the pockets and with very small intensity at the outer parts. We also found that the antiferromagnetic correlations act against the bilayer bonding-antibonding splitting destroying it. The bilayer Fermi surface splitting is practically zero

Cram\'er-Rao bounds for synchronization of rotations

Synchronization of rotations is the problem of estimating a set of rotations R_i in SO(n), i = 1, ..., N, based on noisy measurements of relative rotations R_i R_j^T. This fundamental problem has found many recent applications, most importantly in structural biology. We provide a framework to study synchronization as estimation on Riemannian manifolds for arbitrary n under a large family of noise models. The noise models we address encompass zero-mean isotropic noise, and we develop tools for Gaussian-like as well as heavy-tail types of noise in particular. As a main contribution, we derive the Cram\'er-Rao bounds of synchronization, that is, lower-bounds on the variance of unbiased estimators. We find that these bounds are structured by the pseudoinverse of the measurement graph Laplacian, where edge weights are proportional to measurement quality. We leverage this to provide interpretation in terms of random walks and visualization tools for these bounds in both the anchored and anchor-free scenarios. Similar bounds previously established were limited to rotations in the plane and Gaussian-like noise